**Time-dependent density-functional theory** (**TDDFT**) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photoabsorption spectra.

- Overview
- Formalism
- Runge–Gross theorem
- Time-dependent Kohn–Sham system
- Linear response TDDFT
- Key papers
- Books on TDDFT
- TDDFT codes
- References
- External links

TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system. The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times. Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement.

The formal foundation of TDDFT is the Runge–Gross (RG) theorem (1984)^{ [1] } – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964).^{ [2] } The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density. This implies that the many-body wavefunction, depending upon 3*N* variables, is equivalent to the density, which depends upon only 3, and that all properties of a system can thus be determined from knowledge of the density alone. Unlike in DFT, there is no general minimization principle in time-dependent quantum mechanics. Consequently, the proof of the RG theorem is more involved than the HK theorem.

Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical (interacting) system of interest. As in DFT, this is called the (time-dependent) Kohn–Sham system. This system is formally found as the stationary point of an action functional defined in the Keldysh formalism.^{ [3] }

The most popular application of TDDFT is in the calculation of the energies of excited states of isolated systems and, less commonly, solids. Such calculations are based on the fact that the linear response function – that is, how the electron density changes when the external potential changes – has poles at the exact excitation energies of a system. Such calculations require, in addition to the exchange-correlation potential, the exchange-correlation kernel – the functional derivative of the exchange-correlation potential with respect to the density.^{ [4] }^{ [5] }

The approach of Runge and Gross considers a single-component system in the presence of a time-dependent scalar field for which the Hamiltonian takes the form

where *T* is the kinetic energy operator, *W* the electron-electron interaction, and *V*_{ext}(*t*) the external potential which along with the number of electrons defines the system. Nominally, the external potential contains the electrons' interaction with the nuclei of the system. For non-trivial time-dependence, an additional explicitly time-dependent potential is present which can arise, for example, from a time-dependent electric or magnetic field. The many-body wavefunction evolves according to the time-dependent Schrödinger equation under a single initial condition,

Employing the Schrödinger equation as its starting point, the Runge–Gross theorem shows that at any time, the density uniquely determines the external potential. This is done in two steps:

- Assuming that the external potential can be expanded in a Taylor series about a given time, it is shown that two external potentials differing by more than an additive constant generate different current densities.
- Employing the continuity equation, it is then shown that for finite systems, different current densities correspond to different electron densities.

For a given interaction potential, the RG theorem shows that the external potential uniquely determines the density. The Kohn–Sham approaches chooses a non-interacting system (that for which the interaction potential is zero) in which to form the density that is equal to the interacting system. The advantage of doing so lies in the ease in which non-interacting systems can be solved – the wave function of a non-interacting system can be represented as a Slater determinant of single-particle orbitals, each of which are determined by a single partial differential equation in three variable – and that the kinetic energy of a non-interacting system can be expressed exactly in terms of those orbitals. The problem is thus to determine a potential, denoted as *v*_{s}(**r**,*t*) or *v*_{KS}(**r**,*t*), that determines a non-interacting Hamiltonian, *H*_{s},

which in turn determines a determinantal wave function

which is constructed in terms of a set of *N* orbitals which obey the equation,

and generate a time-dependent density

such that *ρ*_{s} is equal to the density of the interacting system at all times:

Note that in the expression of density above, the summation is over *all* Kohn–Sham orbitals and is the time-dependent occupation number for orbital . If the potential *v*_{s}(**r**,*t*) can be determined, or at the least well-approximated, then the original Schrödinger equation, a single partial differential equation in 3*N* variables, has been replaced by *N* differential equations in 3 dimensions, each differing only in the initial condition.

The problem of determining approximations to the Kohn–Sham potential is challenging. Analogously to DFT, the time-dependent KS potential is decomposed to extract the external potential of the system and the time-dependent Coulomb interaction, *v*_{J}. The remaining component is the exchange-correlation potential:

In their seminal paper, Runge and Gross approached the definition of the KS potential through an action-based argument starting from the Dirac action

Treated as a functional of the wave function, *A*[Ψ], variations of the wave function yield the many-body Schrödinger equation as the stationary point. Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,

and derived a formal expression for the exchange-correlation component of the action, which determines the exchange-correlation potential by functional differentiation. Later it was observed that an approach based on the Dirac action yields paradoxical conclusions when considering the causality of the response functions it generates.^{ [6] } The density response function, the functional derivative of the density with respect to the external potential, should be causal: a change in the potential at a given time can not affect the density at earlier times. The response functions from the Dirac action however are symmetric in time so lack the required causal structure. An approach which does not suffer from this issue was later introduced through an action based on the Keldysh formalism of complex-time path integration. An alternative resolution of the causality paradox through a refinement of the action principle *in real time* has been recently proposed by Vignale.^{ [7] }

Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system. In this case one can analyze the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function so that we can simply use all the properties of DFT.

Consider a small time-dependent external perturbation . This gives

and looking at the linear response of the density

where Here and in the following it is assumed that primed variables are integrated.

Within the linear-response domain, the variation of the Hartree (H) and the exchange-correlation (xc) potential to linear order may be expanded with respect to the density variation

and

Finally, inserting this relation in the response equation for the KS system and comparing the resultant equation with the response equation for the physical system yields the Dyson equation of TDDFT:

From this last equation it is possible to derive the excitation energies of the system, as these are simply the poles of the response function.

Other linear-response approaches include the Casida formalism (an expansion in electron-hole pairs) and the Sternheimer equation (density-functional perturbation theory).

- Hohenberg, P.; Kohn, W. (1964). "Inhomogeneous Electron Gas".
*Physical Review*.**136**(3B): B864. Bibcode:1964PhRv..136..864H. doi: 10.1103/PhysRev.136.B864 . - Runge, Erich; Gross, E. K. U. (1984). "Density-Functional Theory for Time-Dependent Systems".
*Physical Review Letters*.**52**(12): 997. Bibcode:1984PhRvL..52..997R. doi:10.1103/PhysRevLett.52.997.

- M.A.L. Marques; C.A. Ullrich; F. Nogueira; A. Rubio; K. Burke; E.K.U. Gross, eds. (2006).
*Time-Dependent Density Functional Theory*. Springer-Verlag. ISBN 978-3-540-35422-2. - Carsten Ullrich (2012).
*Time-Dependent Density-Functional Theory: Concepts and Applications*. Oxford Graduate Texts. Oxford University Press. ISBN 978-0-19-956302-9.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

**Quantum decoherence** is the loss of quantum coherence, the process in which a system's behaviour changes from that which can be explained by quantum mechanics to that which can be explained by classical mechanics. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

**Density-functional theory** (**DFT**) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

**Electron density** or **electronic density** is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either or . The density is determined, through definition, by the normalised -electron wavefunction which itself depends upon variables. Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a functional to a change in a function on which the functional depends.

The **Schwinger–Dyson equations** (**SDEs**) or **Dyson–Schwinger equations**, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. **Volume charge density** is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. **Surface charge density** (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. **Linear charge density** (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

**Local-density approximations** (**LDA**) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space. Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems.

The **Kohn-Sham equations** are a set of mathematical equations used in quantum mechanics to simplify the complex problem of understanding how electrons behave in atoms and molecules. They introduce fictitious non-interacting electrons and use them to find the most stable arrangement of electrons, which helps scientists understand and predict the properties of matter at the atomic and molecular scale.

In many-body theory, the term **Green's function** is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

The **quantum potential** or **quantum potentiality** is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.

In physics and engineering, **mass flux** is the rate of mass flow. Its SI units are kg m^{−2} s^{−1}. The common symbols are *j*, *J*, *q*, *Q*, *φ*, or Φ, sometimes with subscript *m* to indicate mass is the flowing quantity. Mass flux can also refer to an alternate form of flux in Fick's law that includes the molecular mass, or in Darcy's law that includes the mass density.

In density functional theory (DFT), the **Harris energy functional** is a non-self-consistent approximation to the Kohn–Sham density functional theory. It gives the energy of a combined system as a function of the electronic densities of the isolated parts. The energy of the Harris functional varies much less than the energy of the Kohn–Sham functional as the density moves away from the converged density.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the **Hartree equations** for atoms, using the concept of *self-consistency* that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the **self-consistent field** method.

In quantum mechanics, specifically time-dependent density functional theory, the **Runge–Gross theorem** shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential in which the system evolves and the density of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the electronic density, *ρ*(**r**,*t*) changes in response to an external scalar potential, *v*(**r**,*t*), such as a time-varying electric field.

An **electric dipole transition** is the dominant effect of an interaction of an electron in an atom with the electromagnetic field.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Runge, Erich; Gross, E. K. U. (1984). "Density-Functional Theory for Time-Dependent Systems".
*Physical Review Letters*.**52**(12): 997–1000. Bibcode:1984PhRvL..52..997R. doi:10.1103/PhysRevLett.52.997. - ↑ Hohenberg, P.; Kohn, W. (1964). "Inhomogeneous electron gas" (PDF).
*Phys. Rev*.**136**(3B): B864–B871. Bibcode:1964PhRv..136..864H. doi: 10.1103/PhysRev.136.B864 . - ↑ van Leeuwen, Robert (1998). "Causality and Symmetry in Time-Dependent Density-Functional Theory".
*Physical Review Letters*.**80**(6): 1280–283. Bibcode:1998PhRvL..80.1280V. doi:10.1103/PhysRevLett.80.1280. - ↑ Casida, M. E.; C. Jamorski; F. Bohr; J. Guan; D. R. Salahub (1996). S. P. Karna and A. T. Yeates (ed.).
*Theoretical and Computational Modeling of NLO and Electronic Materials*. Washington, D.C.: ACS Press. p. 145–. - ↑ Petersilka, M.; U. J. Gossmann; E.K.U. Gross (1996). "Excitation Energies from Time-Dependent Density-Functional Theory".
*Physical Review Letters*.**76**(8): 1212–1215. arXiv: cond-mat/0001154 . Bibcode:1996PhRvL..76.1212P. doi:10.1103/PhysRevLett.76.1212. PMID 10061664. - ↑ Gross, E. K. U.; C. A. Ullrich; U. J. Gossman (1995). E. K. U. Gross and R. M. Dreizler (ed.).
*Density Functional Theory*. New York: Plenum Press. ISBN 0-387-51993-9. OL 7446357M. - ↑ Vignale, Giovanni (2008). "Real-time resolution of the causality paradox of time-dependent density-functional theory".
*Physical Review A*.**77**(6): 062511. arXiv: 0803.2727 . Bibcode:2008PhRvA..77f2511V. doi:10.1103/PhysRevA.77.062511. S2CID 118384714.

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